Rabu, 22 Mei 2013

Ruth-Aaron Pair

Whoa, what is Ruth-Aaron Pair?
That is the first thing that crossed my mind when I saw it at wikipedia.

Well, after I read it, I came to this conclusion:

Ruth-Aaron pair consists of two consecutive integers (e.g. 135 and 136) for which the sums of the prime factors of each integer are equal. For example :

77 = 7 x 11
78 = 2 x 3 x 13

and

7 + 11 = 2 + 3 + 13 = 18

If only distinct prime factors are counted, the first few Ruth–Aaron pairs are:
(5, 6), (24, 25), (49, 50), (77, 78), (104, 105), (153, 154), (369, 370), (492, 493), (714, 715), (1682, 1683), (2107, 2108), (2299, 2300), (2600, 2601), (2783, 2784), (5405, 5406), (6556, 6557), (6811, 6812), (8855, 8856), (9800, 9801), (12726, 12727), (13775, 13776), (18655, 18656), (21183, 21184), (24024, 24025), (24432, 24433), (24880, 24881), (25839, 25840), (26642, 26643), (35456, 35457), (40081, 40082), (43680, 43681), (48203, 48204), (48762, 48763), (52554, 52555), (61760, 61761), (63665, 63666), (64232, 64233), (75140, 75141)

If counting repeated prime factor (e.g. 8 = 2 x 2 x 2 and 9 = 3 x 3; 2 + 2 + 2 = 3 + 3), the first few Ruth–Aaron pairs are:
(5, 6), (8, 9), (15, 16), (77, 78), (125, 126), (714, 715), (948, 945), (1330, 1331), (1520, 1521), (1862, 1863), (2491, 2492), (3248, 3249), (4185, 4186), (4191, 4192), (5405, 5406), (5560, 5561), (5959, 5960), (6867, 6868), (8280, 8281), (8463, 8464), (10647, 10648), (12351, 12352), (14587, 14588), (16932, 16933), (17080, 17081), (18490, 18491), (20450, 20451), (24895, 24896), (26642, 26643), (26649, 26650), (28448, 28449), (28809, 28810), (33019, 33020), (37828, 37829), (37881, 37882), (41261, 41262), (42624, 42625), (43215, 43216)

The intersection of the two lists begins:
(5, 6), (7,7, 78), (714, 715), (5405, 5406), (26642, 26643), (52554, 52555), (95709, 95710), (154842, 154843), (173162, 173163), (204258, 204259), (208581, 208582), (248109, 248110), (278277, 278278), (332994, 332995), (417162, 417163), (445305, 445306), (529194, 529195), (554682, 554683), (693610, 693611), (851709, 851710), (869054, 869055), (1232746, 1232747), (1252509, 1252510), (1275546, 1275547), (1275730, 1275731), (1549454, 1549455), (1600962, 1600963), (1607045, 1607046), (1671333, 1671334), (1672710, 1672711), (1777026, 1777027)


Ruth-Aaron Triplets also exist.
If only distinct prime factors are counted :
89460294 = 2 × 3 × 7 × 11 × 23 × 8419
89460295 = 5 × 4201 × 4259
89460296 = 2 × 2 × 2 × 31 × 43 × 8389
2 + 3 + 7 + 11 + 23 + 8419 = 5 + 4201 + 4259 = 2 + 31 + 43 + 8389 = 8465

151165960539 = 3 × 11 × 11 × 83 × 2081 × 2411
151165960540 = 2 × 2 × 5 × 7 × 293 × 1193 × 3089
151165960541 = 23 × 29 × 157 × 359 × 4021
3 + 11 + 83 + 2081 + 2411 = 2 + 5 + 7 + 293 + 1193 + 3089 = 23 + 29 + 157 + 359 + 4021 = 4589

If counting repeated prime factor:
417162 = 2 × 3 × 251 × 277
417163 = 17 × 53 × 463
417164 = 2 × 2 × 11 × 19 × 499
2 + 3 + 251 + 277 = 17 + 53 + 463 = 2 + 2 + 11 + 19 + 499 = 533

6913943284 = 2 × 2 × 37 × 89 × 101 × 5197
6913943285 = 5 × 283 × 1259 × 3881
6913943286 = 2 × 3 × 167 × 2549 × 2707
2 + 2 + 37 + 89 + 101 + 5197 = 5 + 283 + 1259 + 3881 = 2 + 3 + 167 + 2549 + 2707 = 5428

Until 2006, just 4 above triplets are known.

Source : http://en.wikipedia.org/wiki/Ruth%E2%80%93Aaron_pair